Evaluation of center of mass estimation for obese using statically equivalent serial chain

The complex structure of the human body makes its center of mass (CoM) estimation very challenging. The typically used estimation methods usually suffer from large estimation errors when applied to bodies with structural differences. Thus, a reliable estimation method is of utmost importance. In this paper, we present a detailed evaluation of a subject-specific CoM estimation technique named Statically Equivalent Serial Chain (SESC) by investigating its estimation ability over two different groups of subjects (Fit and Obese) in comparison to the segmental analysis method. For this study, we used an IMU-based motion capture system and a force platform to record the joint angles and corresponding center of pressure (CoP) values of twenty-five participants while performing a series of static postures. The root-mean-square errors (RMSE) of SESC’s estimation for both groups showed close and lower mean values, whereas the segmental analysis method showed significantly larger RMSE values in comparison to SESC (p < 0.05). In addition, we used the Bland–Altman analysis to evaluate the agreement between the two techniques and the ground truth CoP, which showed the accuracy, precision, and reliability of SESC over both groups. In contrast, the segmental analysis method did not present neither accurate nor precise estimations, as our analysis revealed considerable fixed and proportional biases.

lacked practicality as it requires the use of force platforms, which renders the experimentation area limited to the small surface of the platform. This constraint is a considerable limitation for studies that require a lot of mobility.
Recent techniques have been introduced to enhance the estimation process, most notably those which relied on machine learning [22][23][24][25][26] . In our previous work 27 , we introduced a method to map the joint angles to the CoM position using a deep neural network (DNN). In that study, the CoM used in the training dataset was collected from 'Fit subjects' using the segmental analysis method, which can be less accurate in cases where the subjects have different body densities as in the case of obesity. This led us to explore a theoretically more accurate and promising technique called statically equivalent serial chain (SESC), proposed by Cotton et al. 28 . This particular technique, based on the work previously done by Espiau and Boulic 29 on CoM estimation in robotics, presents a way to estimate the CoM of a branched chain without the need for inertial parameters. Rather than assigning the anthropometric data to each segment of the body, in this approach, an identification phase has to be carried out before the experimentation phase. This identification phase provides a subject-specific vector of parameters analogically equivalent to the anthropometric data needed to estimate the CoM as in the segmentation method. This 'subject-specificity' feature theoretically gives SESC the ability to overcome the challenges imposed by atypical body shapes in CoM estimations, which led us to further investigate the SESC CoM technique over the obese population. SESC CoM estimation technique had been used and validated for different populations over the years, like the fit, elderly and hemiplegic populations, in different studies [30][31][32] . On the other hand, to our knowledge, its accuracy is not evaluated for the obese population yet. We hypothesize that, theoretically, SESC can overcome the effect of significant body differences, however it still needs to be tested. For this purpose, in this study, we investigated the accuracy of CoM estimations based on the SESC technique over obese subjects, in comparison to its accuracy over fit subjects. Additionally, to show its superiority over the segmental analysis technique we compared our estimation results for both groups, with the estimations based on the segmental analysis technique. To evaluate the estimation performance of those two techniques, in addition to their accuracies, we compared their precision, as well as their agreement with the gold standard reference method.

Methods
Participants. Twenty-five adult individuals were elected to participate in the experiments, eleven fit (BMI < 25 kg/m 2 ) and fourteen obese participants (BMI > 30 kg/m 2 ), with ages ranging between 18 and 30. This study was approved by the Research and Publication Ethics Board of Bahcesehir University and performed at Bahcesehir University Biomedical Engineering Laboratory in Istanbul. All procedures were conformed to the Helsinki Declaration. Informed consent was obtained from all individuals participating in the study. For the joint angles, a low-cost and accurate motion capture device MVN Awinda (XSENS Technologies BV, Enschede, The Netherlands), was used. XSENS has been verified in literature for its reliability and concurrent validity and considered suitable for clinical applications 33 . This system, which operates at a 60 Hz sampling frequency, is comprised of 17 inertial measurement units (IMUs), which transform the human body into a 23 segments biomechanical model with 22 joints. The MVN software version 2020.2 was used.
For the horizontal CoM position, we used the BERTEC FP-4060-05-PT force platform, as the CoP recorded by the force platform is regarded as the horizontal projection of the CoM during static conditions. Center of mass estimation. In this study, we estimated the human CoM using the SESC method and segmental analysis method (provided by XSENS MVN software) during various static conditions, and compared the estimations with the CoP readings recorded by a force platform.
Statically equivalent serial chain. Biomechanical design. The first step in SESC modeling is to design a biomechanical representation of the human body. In this study, a nine-segments nineteen-degrees of freedom (DoF) model is used (Fig. 1). This model is considered to be composed of rigid bodies connected by either spherical (3 DoF) joints representing the shoulders, lumbosacral (L5/S1), and hips, or hinge joints (1 DoF) representing the knees and ankles.
The classical determination of the CoM position of such a structure is computed by calculating the weighted sum of each rigid body's CoM location as described in Eq. (1).
where n is the number of joints, m i is the mass of each segment, C i is the location of each segment's CoM and M is the total mass of the structure M = n i=1 m i . After expanding Eq. (1), it becomes: where the vectors � v 1→9 contain all of the segments' constant parameters ( m i , C i and d i ) and M. After reaching the form of Eq. (3), we observe its resemblance to a serial chain comprised of the links v 1 through v 9 with the CoM located at its end-effector. This end-effector can be located by applying the following matrix multiplication: www.nature.com/scientificreports/ where I is a 3-by-3 identity matrix, R represents the 3-by-3n matrix containing the corresponding joint angles, and V is the 3n-by-1 subject-specific SESC vector.
Identification process. The identification process aims to determine V for each subject. Cotton et al proposed that the SESC vector V can be calculated using the Moore-Penrose pseudoinverse and partial information of the CoM 28 . In this case, the CoP recorded by the force platform is regarded as the ground projection of the CoM during static conditions (hence the term 'statically' in SESC). Using the pseudoinverse, the SESC vector is calculated as follows: Replacing CoM with the CoP values, and removing the vertical axis-related rows from (5) yields the following form: For R′ to be invertible by the pseudoinverse, the number of rows should exceed the number of columns. This requires us to record several distinct joint configurations 'm' , with 'm' being at least equal to 3n/2. Therefore, the equation used to calculate V becomes: SESC estimation. After V is identified, the full 3-dimensional CoM can be calculated with only joint angles values by using Eq. (4). For more detailed explanations about SESC identification, its calculation, and model simplifications please refer to the following studies 30,[34][35][36][37] .
Segmental analysis CoM estimation. The XSENS MVN system used in this study to obtain joint angles, also estimates CoM positions via the segmental analysis method, by first calculating the orientation and position of each body segment and approximating the mass fractions and CoM positions of each segment taking into consideration the anthropometric tables published in 14,38 . The total body CoM is then calculated using the Eq.
(1) previously mentioned in SESC CoM estimation. To compute the segment inertial parameters, such as CoM position and mass fraction, the system uses regression equations and the anthropometric measurements of the subject entered by the user in the "Motion Capture Configuration window". A total number of twelve body measurements are required in order for MVN to approximate its twenty-three segment biomechanical model. Those measurements are: body height, shoulder height, shoulder width, arm span, wrist span, elbow span, hip height, hip width, knee height, ankle height, foot length, and extra shoe sole thickness (considered zero when barefooted).
For segmental analysis CoM estimations in this study, we used the estimations provided by XSENS MVN system directly, by entering the subject's body measurements required by the system, and recording the corresponding CoM location for the various static postures.
Experimental protocol. Calibration phase. Before starting the experiments, the participants were asked to put on a specially designed T-shirt that has specific sensor placement landmarks, Velcro straps, and a headband provided by the Awinda system. These wearables were used to hold in place the sensors which were located on the head, shoulders, sternum, upper arms, lower arms, hands, pelvis, upper legs, lower legs, and feet. The sensor placement was carried out as instructed by the XSENS MVN user manual 39 .
A calibration session was followed afterward, which required the subject to stand still for 2.5 s in an upright position called the N-pose, followed by a 3 meters walk from and to the initial standing position. The calibration process was repeated in case it was evaluated as anything less than "successful" by the MVN software. After the motion capture system is calibrated, the participants were asked to stand on the force platform with each foot placed on the marked locations. The marks placed on the force platform represent specific horizontal points known with respect to the force platform's reference frame. Therefore, by knowing where a subject is standing in both XSENS and the force platform's coordinate systems, we were able to combine both devices in one single www.nature.com/scientificreports/ global frame of reference. These two devices were synchronized using a third-party trigger module (Delsys Trigger module). It should be noted that the data collection was performed for each individual in a single take, and was repeated again from the calibration step in case any problems occurrences.
Static postures and data preprocessing. The SESC identification phase requires a certain number of joint configurations (postures) along with their respective CoP readings 30 , therefore the subjects were asked to perform a series of distinct postures that focused on the movement of the hips, knees, ankles, the L5/S1, and shoulder joints. The maximum number of postures that we requested from each subject was 120 postures, however, the number of selected postures had an average of 103 due to the difficulties of performing some of the postures as they required a certain level of flexibility. The static postures were selected based on the conditions listed in 40 , which states that a posture is considered static if, over a window of 1 second, the standard deviation of the angles measurements is less than 1.5° and the standard deviation of the CoP displacement is less than 6 mm. The recorded postures of each subject were divided into two datasets; one was used to calculate the SESC vector (75% of the postures) and the other (25% of the postures) was used to evaluate the estimated CoM. The root-meansquare error (RMSE) was calculated for the evaluation of the estimated CoM.
Statistical analysis. After calculating the RMSE values for CoM estimations of each subject, the mean, standard deviations and the coefficients of variation of the obtained RMSEs were calculated for each group, and statistical analysis was carried out to investigate the difference in CoM RMSE values between both Fit and Obese groups for both estimation techniques. The analysis was carried out using SPSS (IBM SPSS Statistics 25, IBM Corp., USA) software. The conformity of the data to the normal distribution was evaluated using the Shapiro-Wilk test. To compare the Fit and Obese datasets the Student's t-test was applied in the case of normally distributed data, while the nonparametric Mann-Whitney U test was used for cases that showed a non-normal distribution. For the comparison of SESC and the segmental analysis (XSENS) RMSE results, the paired t-test was used for the case of a normally distributed population while the Wilcoxon signed-rank test was used for the cases that showed a non-normal distribution. The significance level was set as p < 0.05. For further assessment of the results, depending on the normality, either Pearson's correlation coefficient (r) or Spearman's rank correlation coefficient ( r s ) was calculated to quantify the strength of the linear relationship between the estimated and ground truth (CoP) readings. Furthermore, to investigate whether the estimations based on SESC and the segmental analysis are in agreement with the ground truth readings, we used the Bland-Altman analysis 41 in which we calculated the mean difference between the estimated and the ground truth values (fixed bias) and 95% limits of agreemenet (LoA) based on the equation "mean ± 1.96×SD". In addition, a regression analysis was performed to investigate the existence of a proportional bias 42 . For the regression analysis, because we have multiple estimations from the same subjects, we used longitudinal approach 43 .

Results
The CoM of each subject was estimated using SESC's subject-specific vectors, resulting from the identification procedure, and by the segmental analysis method using XSENS' algorithm. The box and whisker plots shown in Fig. 2, present the RMSE values of the CoM estimations based on SESC and XSENS for the Fit and Obese groups. For both groups, along the AP-axis (Fig. 2a), it is clear that the RMSE values belonging to SESC are significantly lower than those of XSENS (p = 0.001). On the other hand, there is no statistically significant difference in the comparison of the RMSE values of SESC estimations for the Fit and Obese Table 2. The RMSEs of the CoM estimations based on SESC and segmental (XSENS) method for 14 Obese subjects along AP and ML axes. The mean and standard deviation (SD) values of the RMSEs of the subjects and the coefficients of variation (CV) are provided in the last three rows.  www.nature.com/scientificreports/ groups (p = 0.788). On the contrary for the XSENS, the RMSE values of the Fit group appear to be significantly higher than those of the Obese group (p < 0.05). The boxplots representing the errors along the ML-axis (Fig. 2b) present different distributions than those of the AP direction. First, by considering the "Fit vs Obese" comparisons, the boxplots show a very close median value (p = 0.57 and p = 0.83 for SESC and XSENS respectively). However, the data distribution of the Obese subjects ("Obese _ SESC") presents a larger IQR than the one of the Fit subjects ("Fit _ SESC"). As for the case of the segmental analysis (XSENS), although the boxplots "Fit _ XSENS" and "Obese _ XSENS" present a few differences regarding the data distribution and skewness, the overall results indicate a good degree of similarity between them. For the SESC and XSENS comparison, from the plots it is visible that for both Fit and Obese groups, SESC had significantly lower RMSE values than those of XSENS (p = 0.005 and p = 0.03 for Fit and Obese groups, respectively).

RMSE-AP (mm) RMSE-ML (mm) RMSE-AP (mm) RMSE-ML (mm)
In Table 3, the results of the Pearson correlation coefficients for the CoP measurements and the CoM estimations by SESC and XSENS are given. Those results show strong correlations between the SESC estimations and the ground truth CoP readings for both groups along both axes. On the other hand, for the XSENS CoM estimations, there is strong correlation along the AP axis (r = 0.82 and r = 0.74 for the Fit and Obese estimations respectively), but weaker correlation along the ML axis (r = 0.55 and r = 0.49 for the Fit and Obese estimations respectively).
For further evaluation of the agreement between the estimation methods and ground truth values, we assessed the Bland-Altman analysis. The analysis results are summarized in Table 4.
For the Fit group, the Bland-Altman plots (Fig. 3) show that the data points based on SESC's estimations (Fig. 3a,b) are scattered close to the zero line with the majority of data points existing within the LoA which ranged from -34.78 mm to 34.4 mm for AP direction (Fig. 3a), and from -34.1 mm to 30.66 mm for the ML direction (Fig. 3b). The mean differences (fixed bias) in the AP and ML directions had values of -0.18 mm and -1.72 mm respectively (Table 4). On the other hand, CoM estimations of XSENS (Fig. 3c,d) present wider ranges of LoA, ranging from -104.04 mm to 30.59 mm (Table 4) in the AP direction (Fig. 3c), and from -63.75 mm to 31.26 mm (Table 4) for the ML direction (Fig. 3d). The fixed bias values are given in Table 4 as -36.72 mm and -16.24 mm in the AP and ML directions respectively.
Furthermore, the scattering of CoM estimations based on XSENS (Fig. 3c,d) shows the existence of a trend described by the 3rd (y = − 9.15 − 0.43x) and 4th (y=1.82-0.46x) equations presented in Table 4.
For the Obese group, similar to the results of the Fit group, the Bland−Altman plots (Fig. 4) show that the data points related to SESC's estimations (Fig. 4a,b) are also scattered close to the zero line with the majority of data points existing within the LoA which ranged from − 41.12 to 40.33 mm for AP direction, and from − 40.57 to 31.36 mm for the ML direction. The mean differences (fixed biases) for the AP and ML directions had values of − 0.39 mm and − 4.60 mm respectively (Table 4).
On the other hand, XSENS' data ( Fig. 4c,d)   www.nature.com/scientificreports/ the fixed biases were − 21.15 mm and − 15.32 mm in the AP and ML directions respectively. Furthermore, the scattering of the data points in Fig. 4c and d again show the existence of a trend.

Discussion
Given the importance of having a reliable CoM estimation technique that can overcome the limitations of the regularly used methods, whether it is being restricted by insufficient space as in the case of using a force platform, or being vulnerable to different body types and weights like the segmental method, SESC is theoretically a strong candidate for CoM estimation. In this study, we conduct an evaluation of the SESC method, in comparison to the segmental analysis method, covering their accuracy, precision, and the ability to maintain a consistent estimation when applied to subjects with significant body differences via evaluating their estimations for Fit and Obese subjects. Considering the SESC RMSE results, the values of the mean errors shown in Tables 1 and 2 denote a good CoM estimation error for SESC when applied to both Fit and Obese groups. The mean RMSE value along the AP-axis was 18.20 (± 6.97) mm for the Obese group, which is close to the mean RMSE value of the Fit group (18.95 ± 6.64) (p = 0.822), thus indicating that along this axis, SESC's estimation ability was not affected by the change of body structure.
On the other hand, the mean RMSE value along the ML-axis was higher by 6.8 mm for the Obese group. However, this difference is not statistically significant (p = 0.57), therefore it shows that the body type change had not affected SESC's estimation error along this axis as well. Furthermore, it should be noted that the SESC RMSE values along the ML-axis did show a wider dispersion of data for Obese group and a considerably larger SD with respect to the mean (CV = 65.19%), however, in order to properly interpret those results, a larger number of samples should be investigated in addition to a wider diversity of body-weights among the groups. www.nature.com/scientificreports/ As for XSENS' results, along the AP-axis, the RMSE values are considerably high for both Fit and Obese groups (average RMSE values are 51.47 mm and 38.52 mm, respectively). Although the average error for the Obese group was significantly less than for the Fit group (p < 0.05), it is, on the other hand, still significantly higher than SESC's RMSE values (p = 0.001). As for the ML-axis, XSENS' RMSE values showed no statistically significant difference between both groups (p = 0.837).
By comparing our SESC RMSE values for both groups to the results presented by González et al. 30 , in which they compared SESC and the Antropometric-based estimations using a gold standard motion capture system (Vicon) and a low-cost device (Kinect), we can determine that our SESC estimations are fairly acceptable for both groups. According to that study, the average SESC-CoM RMSE values ranged from 10.2 to 23.37 mm using both devices, which is comparable to our SESC results, as they fall within the same range.
Regarding the linear correlation between the estimated CoM and the CoP measurements, Table 3 shows that there exists a strong correlation between the SESC estimations and ground truth CoP for both groups in both AP and ML axes. On the other hand, the XSENS estimations show a strong correlation along the AP-axis, and a moderate correlation along the ML-axis. As a result of the strong correlation between the estimation techniques and the ground truth, we conduct a deeper evaluation of those techniques in terms of accuracy and precision.
Using the Bland-Altman method, we further evaluated the agreement between the estimations of each method with the ground truth in both AP and ML axes. According to the plots shown in Fig. 3, we determined that the SESC results for the Fit group (Fig. 3a,b) are in agreement with the ground truth measurements, as the data points are mostly spread within the LoA with no indicator of either fixed or proportional biases. Furthermore, both plots (Fig. 3a,b) show that the mean difference lines are close to '0' indicating a good accuracy in estimation 42 . In addition, these plots also show that the measurements are spread close to each other and the LoA are close to the bias line, thus implying that this technique is precise as well 42 .
On the other hand, Fig. 3c and d, representing the cases for XSENS estimations, show that these values do not agree well with the ground truth readings, as the LoA had larger ranges and CI values than those of SESC's www.nature.com/scientificreports/ case ( Table 4). The scattering of the data in those plots also indicates a lack of accuracy, as the data points are spread far from the '0' value. Furthermore, we also interpret that there exists a lack of precision with XSENS' estimations, as the data points scatter away from each other, in contrast to what was seen in the case of SESC. In addition, XSENS' results clearly present fixed and proportional biases along both AP and ML axes. According to Table 4, the fixed biases (-36.72 mm and -16.24 mm for AP and ML axes, respectively) were significantly large in comparison to SESC results (-0.18 mm and -1.72 mm for AP and ML axes, respectively). Regarding the proportional bias, Table 4 shows that regression models in XSENS's case have significantly larger R 2 values (0.430 and 0.476 for the AP and ML axes, respectively) than SESC (0.012 and 0.075 for the AP and ML axes, respectively), which confirms the existence of a proportional bias in XSENS.
Regarding the Bland-Altman analysis of the CoM estimations for the Obese group, the SESC estimations (Fig. 4a,b) show a good agreement with the ground truth values, similar to the results achieved with the Fit group (Fig. 3a,b), which also confirms the accuracy and precision of SESC. Hence, the general evaluation of the SESC estimations strongly denotes that this method can consistently provide an accurate and precise CoM estimation for humans with different body mass distributions. Nevertheless, we should point out that a slight trend was detected in the plot shown in Fig. 4b indicating a sign of a small proportional bias ( R 2 = 0.185) with SESC's estimations along the ML axis. This requires further investigations with a larger number of subjects and more variety in BMI especially for the Obese group. As for the segmental method, similar to its results on the Fit group, it still showed an absence of agreement between its estimations and the ground truth values (Fig. 4c,d), in addition to the existence of large fixed biases (− 21.15 mm and − 15.32 mm for AP and ML axes, respectively), and clear indications of proportional biases ( R 2 = 0.320 and R 2 = 0.384 for AP and ML axes, respectively).
Although the findings of this study give a valuable evaluation of SESC, several limitations should be taken into consideration. First, in this study, we only investigated the CoM estimation along the AP and ML axes without referring to the vertical CoM estimation, as there is still no gold standard method available in the literature, agreed upon by the researchers, that can be used as reference. Second, the evaluation of the CoM in this study is only addressed in static conditions, which is also a considerable limitation that should be further investigated for a better validation of SESC's estimation capabilities. Additionally, although the number of subjects used in this study was reasonably acceptable, an increase in the number of participants will give a more robust interpretation of the results. Moreover, increasing the biomechanical architecture's complexity (i.e., increasing the number of segments and DoFs) can further improve the accuracy of the SESC's CoM estimation. Lastly, our future studies will focus on evaluating the CoM estimation using both SESC and the segmental analysis method over several time periods in order to investigate their test re-test reliability.

Conclusion
In this paper, we investigated SESC's CoM estimation ability in overcoming body differences in comparison to the segmental method, by applying it to two groups (Fit and Obese) with different body mass indexes. The SESC's estimation results were comparable to those mentioned in the literature for both groups and showed complete superiority over the estimation results of the segmental analysis method. In conclusion, we determined that SESC's accuracy and precision were consistent regardless of the structure of the body under study, and its CoM estimation is suitable for balance assessment for the populations such as the Obese. As for the segmental method, adopted by XSENS software, the results showed significantly larger errors than SESC and a lack of accuracy and precision in CoM estimation for both groups along the AP and ML axes. Further investigations could be carried out to deal with the biases, and offer a better calibration equation to fix the estimation errors.